Borehole sources, including chemical explosives, air gun, water gun, and piezoelectric transducers in the borehole, generate seismic waves inside and outside the borehole. Modeling the wavefield is of key importance in acoustic logging, crosshole tomography, mining geophysics, and deep sounding seismic for interpretation of amplitude information of real data and prediction of energy-radiation patterns. Classic methods for modeling the wavefield inside a borehole, such as real-axis integration, are challenged by highly oscillatory integrals encountered when modeling the wavefield outside the borehole. We have developed a novel method, called steepest descent integration (SDI), which evaluates the oscillatory wavenumber integration by numerically integrating along the steepest descent path. The oscillation along the new integration path is significantly reduced. The contributions of poles and branch cuts are added if they are located between the steepest descent path and the real axis. The SDI is applicable to arbitrary frequency and source-receiver distance. Comparison with real-axis integration shows that the method can compute highly oscillatory integrals with better efficiency and accuracy. In addition, the SDI is more numerically robust because it generates no spurious arrivals, which are evident in the real-axis integration. Analysis of numerical examples at different source-receiver distance shows that SDI is more efficient when computing far-field seismograms. This SDI can also be used to compute highly oscillatory integral in other wave-propagation problems.
Numerical approximation of a Tikhonov type regularizer by a discretized frozen steepest descent method
